Hölder stability estimates in determining the time-dependent coefficients of the heat equation from the Cauchy data set

Proof of Lemma 2.5.

Define the space

We consider the subspace 𝒜 T of L 2 ⁢ ( 0 , T ; H λ - 1 ⁢ ( ℝ n ) ) defined by 𝒜 T = { 𝒫 φ * ⁢ z : z ∈ ℋ T } and let v ∈ L 2 ⁢ ( Q ) . Henceforth, 〈 ⋅ , ⋅ 〉 denotes the scalar product in L 2 ⁢ ( ℝ n + 1 ) . We consider the linear operator ℓ on the linear subspace 𝒜 T of L 2 ⁢ ( 0 , T ; H λ - 1 ⁢ ( ℝ n ) ) as follows:

By Lemma 2.3

holds for z ∈ C 1 ⁢ ( 0 , T ; C 0 ∞ ⁢ ( Ω ) ) satisfying z ⁢ ( T , x ) = 0 .

We deduce ∥ ℓ ∥ ≤ C ⁢ ∥ v ∥ L 2 ⁢ ( Q ) . Thus ℓ is continuous in the subspace 𝒜 T of the Hilbert space L 2 ⁢ ( 0 , T ; H λ - 1 ⁢ ( ℝ n ) ) . Therefore, using the Hahn–Banach theorem, we can extend ℓ to a linear function, which we also denote by ℓ , with the same norm to the space L 2 ⁢ ( 0 , T ; H λ - 1 ⁢ ( ℝ n ) ) . By the Riesz representation theorem, there exists a unique solution u ∈ L 2 ⁢ ( 0 , T ; H λ 1 ⁢ ( ℝ n ) ) such that ℓ ⁢ ( f ) = 〈 f , u 〉 for f ∈ L 2 ⁢ ( 0 , T ; H λ - 1 ⁢ ( ℝ n ) ) , with

Choosing f = 𝒫 φ * ⁢ z , we obtain

that is, 𝒫 φ ⁢ u = v .

Since v ∈ L 2 ⁢ ( Q ) and u ∈ L 2 ⁢ ( 0 , T , H 1 ⁢ ( Ω ) ) , using the expression of 𝒫 φ , we deduce that ∂ t ⁡ u ∈ L 2 ⁢ ( 0 , T ; H - 1 ⁢ ( Ω ) ) . Then

On the other hand, from (A.3) we have

Since 𝒫 φ ⁢ u = v , we have

Integrating by parts, we obtain that

holds for every z ∈ C 1 ⁢ ( 0 , T ; C 0 ∞ ⁢ ( Ω ) ) such that z ⁢ ( T , x ) = 0 . Hence we conclude that u ⁢ ( 0 , x ) = 0 in Ω. ∎